steinmetz cp complex-quantities-and-their-use-in-electrical-engineering

ÆTHERFORCE
COMPLEX QUANTITIES
AND
THEIR
USE IN
ELECTRICAL ENGINEERING.
BY CHA8. PROTEUS
8TEINMKTZ.
I.—INTRODUCTION.
In
the
following,
I
shall outline
a
method
of
calculating alter-
nate current phenomena, which,
I
believe, differs from former
methods essentially
in
so
far,
as
it
allows us
to
represent
the

alter-
nate current,
the
sine-function
of
time,
by a
constant numerical
quantity,
and
thereby eliminates
the
independent variable "time"
altogether
from
the
calculation
of
alternate current phenomena.
Herefrom results
a
considerable
simplification
of
methods.
Where before
we had to
deal with periodic functions
of an in-
dependent variable, time,

we
have
now to add,
subtract,
etc.,
constant
quantities—a
matter
of
elementary
algebra—while
problems like
the
discussion
of
circuits containing distributed
capacity, which before involved
the
integration
of
differential
equations containing
two
independent
variables:
"
time
" and
" distance,"
are now

reduced
to a
differential equation with
one
independent variable only,
"
distance," which
can
easily
be in-
tegrated
in its
most general
form.
Even
the
restriction
to
sine-waves, incident
to
this method,
is
no limitation, since
we can
reconstruct
in the
usual
way the
com-
plex harmonic wave from

its
component
d
ine-waves;
though
al-
most always
the
assumption
of the
alternate current
as a
true
sine-wave
is
warranted
by
practical experience,
and
only under
rather exceptional circumstances
the
higher harmonics become
noticeable.
In
the
graphical treatment
of
alternate current phenomena
different representations have been used.

It is a
remarkable
fact, however, that
the
simplest graphical representation
of
ÆTHERFORCE
84
8TEINMETZ
ON COMPLEX QUANTITIES.
periodic functions, the common, well-known polar coordinates;
with time as angle or amplitude, and the instantaneous values of
the function as radii
vectores,
which has proved its usefulness
through centuries in other branches of science, and which is
known to every mechanical engineer
from
the Zeuner diagram
of valve
motioL
of the
Steele,
and
shoald eonse^Tflv
be known to every electrical engineer also, it is remarkable that
this polar diagram has been utterly neglected, and even where it
has been used, it has been misunderstood, and the
sine-wave
rep-

resented—instead of by one
circle—by
two circles, whereby
the phase of the wave becomes indefinite, and hence the diagram
Fio.
l.
useless. In its place diagrams have been proposed, where re-
volving lines represent the instantaneous values by their projec-
tions upon a fixed line, etc., which diagrams evidently are not
able to give as plain and intelligible a conception of the varia-
tion of instantaneous values, as a curve with the instantaneous
values as radii, and the
time
as angle. It is easy to understand
then, that graphical calculations of alternate current phenomena
have found almost no entrance yet into the engineering practice.
In graphical representations of alternate currents, we shall
make use, therefore, of the Polar Coordinate System,
repre-
senting the time by the angle
tp
as amplitude, counting from an
ÆTHERFORCE
8TEWMETZ
ON COMPLEX QUANTITIES. 35
initial radius o A chosen as zero time or starting point, in posi-
tive direction or
counter-clockwise,*
and representing the time of
one complete period by one complete revolution or 360° = 2 n.

The instantaneous values of the periodic function are repre-
sented by the length of the radii
vectores o
B
=
r,
correspond-
ing to the different angles
<p
or times
t,
and every periodic
function
is
hereby represented by a closed curve (Fig. 1).
At
any
time t, represented by angle or amplitude
<p,
the instantaneous
value of the periodic function is cut out on the movable radius
by its intersection o
B
with the characteristic curve c of the
func-
FIG.
2.
tion, and is positive, if in the direction of the radius, negative,
if in opposition.
The

sme^wave
is represented by one circle (Fig. 2).
The diameter o c of the circle, which represents the sine-wave,
ia
called the intensity of the
sine-wave,
and its amplitude,
A
O
B =
«5,
is called the phase of the sine-wave.
The sine-wave is completely determined and characterized by
intensity
and phase.
It is obvious, that
the
phase is of interest only as difference of
phase,
where several waves of different phases are under con-
sideration.
•This
direction of rotation
has been
chosen
as
positive,
since it
is the
direc-

tion of rotation of celestial bodies.
ÆTHERFORCE
86
8TEINMETZ ON COMPLEX QUANTITIES.
"Where
only the integral value* of the sine-wave, and not its
instantaneous values are
required,
the characteristic circle c of
the
sine-wave
can be
dropped,
and its diameter o c considered as
the representatation of the
sine-wave
in the
polar-diagram,
and
in this case we can go a step
farther,
and instead of using the
maximum value of the wave as its representation, use the
efect-
, i
•
i
• ,1 • •
maximum value
we

value,
which
in
the
Bine
wave
is
=
;=
Where, however, the characteristic circle is drawn with the
effective
value as diameter, the instantaneous values, when taken
from the diagram, have to be enlarged by
4/2.
FIG. 8.
We see
herefrom,
that:
11
In polar coordinates, the
sine-wave
is represented in in-
tensity and phase by a vector o c, and in combining or dis-
solving
sine-wa/aes,
they are to be combined or dissolved by the
parallelogram or polygon of
sine-waves?
For the purpose of calculation, the sine-wave is represented
by two

constants:
C,
a>,
intensity and phase.
In this case the combination of sine-waves by the Law of
Parallelogram, involves the use of trigonometric functions.
The
sine-wave
can be represented also by its rectangular
co-
ordinates, a and b (Fig. 3),
where:
ÆTHERFORCE
8TEINMETZ ON COMPLEX QUANTITIES. 87
a = C cos &
b = C sin to
Here a and
J
are the two rectangular components of the
sine-
wave.
This representation of the sine-waves by their rectangular
components a and b is very useful in so
far as
it avoids the use of
trigonometric
functions.
To combine sine-waves, we have sim-
ply to add or subtract their rectangular components. For
instance, if a and b are the rectangular components of one sine-

wave,
a
x
and
b
l
those of another, the resultant or combined sine-
wave has the rectangular components a
-{-
a
1
and b
-\-
b
l
.
To distinguish the horizontal and the vertical components of
sine-waves, so as not to mix them up in a calculation of any
greater length, we may mark the ones, for instance, the vertical
components, by a distinguishing index, as for instance, by the
addition of the letter
j,
and may thus represent the
sine-wave
by
the
expression:
a+jb
which means, that a is the horizontal, b the vertical component
of the

sine-wave,
and both are combined to the resultant
wave:
which has the
phase:
tan
<3
=
a
Analogous, a
—j
b means a sine-wave with a as horizontal,
and — b as vertical
component,
etc.
For the
first,,;"
is nothing but a distinguishing index without
numerical
meaning.
A wave, differing in phase from the wave a
-\-j
b by 180°, or
one-half period, is represented in polar coordinates by a vector
of opposite direction, hence denoted by the algebraic
expression:
— a
~j
b.
This

means:
"
Multiplying the algebraic expression a
+
j b
of
the sine-
wave by — 1, means reversing the
wave,
or rotating it by 180° =
one-half
period.
A wave of equal strength, but lagging 90° =
one-quarter
period behind a
-f-
j
J,
has the horizontal component — b, and
ÆTHERFORCE
1
88
STEINMETZ
ON COMPLEX QUANTITIES.
the vertical component a, hence is represented algebraically by
the
symbol:
j a — b.
Multiplying,
however:

a -\-jb
byj,
we get:
ja+fb
hence, if we define
the—until
now
meaningless—symbol
j so, as
to say,
that:
hence:
' j (a
-\-j
b) = j a — b,
we have :
" Multipling
the
algebraic
expression
a-\-jbofthe
sine-wave
by
j,
means rotating the wave by 90°, or
one-quarter
period,
that
is,
retarding the wave by

one-quarter
period."
In the same
way:
"
Multiplying by —
j,
means advancing the wave by one-
quarter
period"
f
= — 1
means:
j =
^—
i,
that is:
u
j
is the imaginary unit, and the
sine-wave
is represented by a
complex
imaginary
quantity a
-\-j
b."
Herefrom
we get the
result:

"
In the polar diagram of time, the
sine-wave
is represented
m
intensity as well
as phase
by one complex quantity:
*
+j
h
where
a
is the horizontal, b the vertical component of the wave,
the intensity is given by; C =
Va*
-\- ¥
and the phase by: tan at = -,
and
it is; a =
Ccos
at
b = C
s\n
to
hence the wave: a
-\-j b
can also be
expressed
by:

C (cos at
-\-
j sin
at)."
Since we have seen that
sine-waves
are combined by adding
their rectangular
components,
we have :
" Sine-waves
are
combined
by adding their complex algebraic
expressions."
For instance, the
sine-waves:
a
+jb
and a
1
-\-j
b
l
ÆTHERFORCE
STEINMETZ
ON COMPLEX QUANTITIES.
89
combined give the
wave:

A+jB
=
{a
+
a
1
)
+j (b
+
b
l
).
As seen, the combination
of
sine-waves
is
reduced hereby to
the elementary algebra of complex quantities.
If C
=
o
-f-/
c
1
is a sine-wave
of
alternate current, and
r is
the resistance, the
K.

M.
F.
consumed by the resistance is in phase
with the current, and equal
to
current times resistance, hence
it is:
rC=rc-\-jrc
L
.
If
L is
the
"
coefficient
of
self-induction," or
*
=
2
it N
L
the
"
inductive
resistance"
or
" ohmic
inductance," which
in

the following shall be called the
"
inductance," the
E.
M.
F.
pro-
duced
by
the inductance (counter
E.
M.
F.
of self-induction)
is
equal to current times inductance, and lags 90° behind the cur-
rent, hence it is represented by the algebraic
expression:
jsC
and the
E.
M.
F. required
to
overcome the inductance
is
conse-
quently
:
—j s C

that
is,
90° ahead of the current (or, in the usual expression, the
current lags 90° behind the
E.
M.
F.).
Hence, the
E.
M.
F.
required to overcome the resistance
r
and
the inductance
*
is
:
ir-js)C
that
is:
" / =
r
—j s
is the
expression
of
the impedance,
in
complex

quantities,
where
r
— resistance,
s
=
2
n
iV"
L
=
inductance."
Hence,
if
C = c
+
j
c
1
is the current, the
E.
M.
F. required to
overcome the impedance
I
=
r
—j
s
is:

E
— IC
=
(r —j
s)
(c
-\-j
o
x
),
hence,
since,;"
8
=
—
1:
=
(r c
-{-
8
c
1
)
-\-j (r
c
l
—
*
c)
or,

if E =
e
-\-j
e
x
is the impressed
E.
M.
F.,
and
I
=
r
—,;" *
is
the impedance, the current flowing through the circuit
is:
E
e+je*
1
r—js
or, multiplying numerator and denominator by
(r +
j
s),
to elim-
inate the imaginary from the
denominator:
G
—

r*
+
s*
""
r
2
+
«
2
+ •?
r*
-f
s
2
ÆTHERFORCE
40 8TEINMETZ ON COMPLEX QUANTITIES.
If
K\B
the capacity of a condenser, connected in series into
a circuit of current C — c
-\-j
c\
the
B.
M.
F.
impressed upon
the.terminals
of the condenser is E
—

r^j^,
and lags 90°
behind the
current,
hence represented by :
C
where k =
„—^-^-can
be called the
"
capacity inductance
"
or simply
"inductance"
of the condenser. Capacity induc-
tance is of opposite sign to magnetic inductance.
That means:
"
If r = resistance,
L —
coefficient
of
self4iiduction,
hence s = 2
it N
L —
%n-
ductance,
K ==
capacity,

hence
k =
o
ff
\r y
= capacity inductance,
I=r
—j
(s — k) is the
impedance
of
the
circuit,
and Ohm's
law is
re-established
:
E= I
C,
c
=
/ =
E
T
E
in a more
general
form,
however,
giving not only the intensity,

but also
the
phase of the
sime-wowes,
hy their
expression
in
com-
plex
quantities."
In the following we shall outline the application of complex
quantities
to various problems of alternate and polyphase cur-
rents,
and shall show that these complex quantities can be ope-
rated upon like ordinary algebraic numbers, so that for the solu-
tion of most of the problems of alternate and polyphase cur-
rents,
elementary algebra is sufficient.
Algebraic operations with complex quantities:
f
=
-
1
a
-j-
j b
=
c (cos
ai

-{-
j sin
w)
c =
Va*
+
b\
.
b
tan to
•=
a
ÆTHERFORCE
8TBINMBTZ
ON
COMPLEX
QlTANTlTTfiS.
41
If
a -\-
j b =
a
1
-\-j
b\
it must be: a =
a\
b = ft
1
.

Addition and
subtraction:
(«
+J
*)
±
(*
l
+i
ft
1
)
=
(•
±
«*)
+i
(*
± ft
1
)-
Multiplication:
(*
+i
ft)
(«
!
+i
ft
1

) =
(«
<*
x
—
ft
ft
1
)
+i
(«
ft
1
+ ft
«')•
Division:
a+jb
_
i*+jl>){*
x
—jV)
_
act +
hh
1
,
.a
l
b
—

ab
l
a
l
+jb
l
~
a
18
+
ft
12
-
a
1%
-\-
J
18
+^
a
l8
-f-
b
ir
Difference of phase
between:
a
-\~
j b
=

c (cos
at -{-j
an.
<3)
and,
a
1
-f-y
J
1
=
c
1
(cos a*
1
-j-^*
sin
a*
1
):
?
_
*
tan
IM
1
— tan
to
a
1

a
_
aft
1
—
ft ft'
ten(cs'-«)=
1+teii<3tan<51
=—^-«.t
+
}y
Multiplication by — 1 means reversion, or rotation by 180° =
one-half
period.
Multiplication by j means rotation by 90°, or retardation by
one-quarter
period.
Multiplication by
—,;"
means rotation by
—
90°, or advance
by one-quarter period.
Multiplication by cos
a> -\-j
sin
id
means rotation by angle to.
II.
CIECUITS

CONTAINING RESISTANCE, INDUCTANCE AND
CAPACITY.
Having now established Ohm's law as the fundamental law of
alternate currents, in its complex form :
E=
IC,
where it represents not only the
intensity,
but the phase of the
electric quantities
also,
we can by simple application of Ohm's law
—in
the same way as in continuous current circuits, keeping in
mind,
however, that
JS",
C,
I are complex
quantities—dissolve
and
calculate any alternate current circuit, or network of circuits,
containing resistance, inductance, or capacity in any combination,
without meeting with greater difficulties than are met with in
continuous current circuits. Indeed, the continuous current dis-
tribution appears as a particular case of the general problem,
characterized by the disappearance of all imaginary terms.
As an instance, we shall apply this method to an
inductive
ÆTHERFORCE

42
8TELVMETZ
ON COMPLEX QUANTITIES.
circuit,
shunted
by a
condenser, and
fed
through inductive
mains, upon which a constant alternate
E.
M.
F. is impressed, as
shown
diagrammatically
in Fig. 4.
Let
r =
resistance,
L
=
coefficient of self-induction, hence
8
= 2
it JV
L
— inductance,
and:
/
= r

—j 8
=
impedance of consumer circuit
Let
T*I
=
resistance of condenser leads,
K=
capacity, hence
k = Xrrjy
=
capacity inductance,
and:
2
it Jy
K.
l
i
=
r
x
-\-j
k
=
impedance of condenser circuit.
Letr
0
=
resistance,
Z

0
=
coefficient of self-induction, hence
s
0
=
2it
NL
9
=
inductance, and:
TtW
dm
Fie.
4.
/
Q
=
r j
So
— impedance of the two main
leads.
Let^
=
E.
M.
F.
impressed upon the circuit.
We have then,
if, E =

E.
M.
F.
at ends
of
main
leads,
or at
terminals of consumer and condenser
circuit:
Current
in
consumer circuit, C
=
Current in condenser
circuit,
C
x
=
Hence, total
current,
O
0
—
0 +
C,
=
E\-j
+
-j-J

E
I
K»
M
;.p.
consumed
in
main leads
E
1
=
C
0
1
0
=
£
y-j-
+
-j*
/
Hence, total
E.
M.
F.
E*
^E-\-E
l
=E j1
4-y+y

|
ÆTHERFORCE
STEfNMETZ
ON COMPLEX
QUANflTMS.
48
E
II
or,
E.
M.
F.
at end of main leads, E =
T
—
7
f
*— y-
/
0
I
+
-*o •* 1
+
111
K.
M.
F.
consumed
by

main
leads,
E
l
=
/V V
V~,
I
r
la
I
-\- 1$
l\-\-
11\
E
EI
Current in consumer circuit, C =
—=.
= °
!
/
/
0
/+
/.A+
/
I
x
E
EI'

Current in condenser
circuit,
C
x
=
-^
=
—,,—^i
J-J
I\
I a
I+
loli + Hi
Total current,
<7
C
=
C
+
Ci
=
f?^^^
TI
I
0
I-\-I
0
I\-\-II\
Substituting herein the values,
J-t,

=
**o
.7
*0
I = r—j
8
and,
I
0
I
-\-1
0
Ii -\~
11\
=
a,
—j
b,
where,
a =
r
0
r +
r
0
r,
-f-
r
r
t

—
8
0
8
+
8
0
k
-\-
s k
5
=
*
0
r +
*
0
^i
+
*
r,
+ * /•„
—
r
Q
k
— r
k
we get
E

(
.£*
=
, '
j
[p^rr^ak^MriB—r#)]+.?[^(
rr
i+*£)—afo*—»"£)]
[•
^
=
^p |
(*i
*
- *
*)
+ j
(r,
ft
+ * a)
|
ffi=£^{('«+«*)+./>*-*«)}
As an instance, we may consider the
case:
E
0
= 100 volts,
r
9
= 1 ohm

) r
= 2 ohm
)
r
x
= 0
)
*.
= 10 ohm
) *
= 10 ohm
j
k = 20 ohm
)
/
0
= 1
—10.;
7=2
—
10.;
Ii
=
20j
hence
a
= 302
J
= — 30
Substituting these values, we get,

E
0
= 100,
E = 68.0 (.98 +
XI
$),
E
1
= 35.1 (.94 — .34,;),
C = 6.6 (.10 +
.99/"),
ÆTHERFORCE
44
8TEINMETZ
ON COMPLEX
QUANTITlEB.
O
t
= 3.4 (.17 —
.98
J),
C
0
=
3.4 (.37 +
.93
j),
where the complex quantities are represented in the form c
(cos
a>

+ j sin
<3),
so that the numerical value in
front
of the paren-
thesis gives the
effective
intensity, the parenthesis gives the phase
of the alternate current or
E.
M.
F.
This means: Of the 100 volts impressed, 35.1 volts are con-
sumed by the leads, and 68.0 volts left at the end of the line.
The main current of 3.4 amperes divides into the consumer
current of 6.6 amperes, and the condenser current of 3.4
amperes.
Increasing,
however, the capacity
JST,
that is reducing the capac-
ity inductance to
k
= 10, or
I
y
=
10
j,
we

get:
a = 102,
b =0.
Hence:
E
0
= 100,
E = 100 (.98 +
.20
J),
E
l
= 19.9 (.10
—.99^),
C
=9.8jf,
C
x
= 10 (.20 —
.98
j),
C
0
= 1.98.
Here, though the leads consume 19.9 volts, still the full
potential of 100 volts is left at their end.
1.98 amperes in the main line divide into two branch currents,
of 9.8 and of 10 amperes. We have here one of the frequent
cases,
where one alternate current divides into two branches, so

that either branch current is larger than the undivided or total
current.
Increasing the capacity still further to
*
= 5, or
/,
=
5
j,
gives:
a = 2,
b = 15.
Hence:
E
0
= 100,
E =337 (.32
—.95^),
^i
=
318
(—.03
+j\
G =33.0 (.99
+1SJ),
Ci =
96.3 (1 + .06
J).
C
0

= 63.6(1 +
.03./),
That means, in the leads self-induction consumes an E.
M.
F. of
318 volts, and still 337 volts exist at the end of the line, giving
ÆTHERFORCE
STBINMETZ
ON COMPLEX QUANTITIES. 45
a rise of potential in the leads of 237 volts, due to the combined
effect of self-induction and capacity.
The main current of 63.6 amperes divides into the two branch
currents of 33.0 and 96.3 amperes.
The current which passes over the line is far larger than the
current which in the absence of capacity would be permitted
by
the dead resistance of the line. While in this case 63.6 amperes
flow over the line, a continuous
E.
M.
F.
of 100 volts would send
on
ty
'-
A_
„ = 33.3
amperes over the
line;
and with an

alter-
'0
1'
nating
E.
M.
F.,
but without capacity the current would be limited
to 4.95 amperes only, since in this
case:
C =
,
J^ ,
r—v
=
,
1(
H
•
=
4.95
(.15
+
.99/).
•
(r.
+
r)—,;(^
+ «)
8

—20,;
V T J)
Even by
shorteircuiting
the line, we get
only:
a,
= ^^
o
=
r
^=10(.l + .99A
or 10 amperes over the line.
Hence we have in this arrangement of a condenser shunted to
the inductive circuit and fed by inductive mains, the curious re-
sult that a
short-circuit
at the terminals of the consumer circuit
reduces the line current to about one-sixth.
As a further instance, we may consider the problem :
" Wh(t
is the maximum power which can be transmitted over
an inductive line into a
non-inductwe
resistance, as
lights,
and
how far can this output be increased by the
uie
of shunted

capacity."
Let,
r
0
= resistance,
s
0
= inductance,
hence,
I
0
=
r
0
—j
s
Q
=
impedance
of the line.
Let r = resistance of the consumer circuit, which is shunted
by the capacity inductance k.
r and k are to be determined as to make the power in the
receiving
circuit:
C
2
r, a maximum.
In a continuous current circuit the maximum output is
reached, if r =

r
0J
or E =
-
£-,
where
E
0
is the
E.
M.
F. at the
beginning,
K
the
E.
M.
F.
at the end of the line, and C =
a
hence:
»«
ÆTHERFORCE
46
8TBINMETZ
ON COMPLEX QUANTITIES.
P =
j-
2
-

the maximum output at
efficiency
50 per cent
4:
r
0
Hence, if
E
0
= 100,
r
0
= 1, it
is:
P = 2,500 watts.
Very much less is the maximum output of an alternate
cur-
rent circuit. With an alternate E.
M.
F.
E
m
but without the use
of a condenser, the impedance of the whole circuit
is:
1
—
r
a
+ r —j

s
m
u
*i.
*
n
E
* Eo{ro + r — j
s
0
)
hence the
current:
V =
-j-
=
—j-—;—x*
,
\ %
1
{r
0
-f-
ry +
s
0
E
0
l
r

0
+ r
.
.
*
0
j
r
°
+ r
i
•
I
^0+^
+ ^
\
V(r
0
+
rf
+
*
0
3 1 J
f'fo
+
r)*
+
*
0

M
the
E.
M.
F.
at end of line :
w—c
_
E
»
T
i
r
Q
+ r
.
___
s
0
l
T
•(r.+^H?
\
f(r.+rJ«+#
-
» "•"'
•(• +#•}»+«.*
J '
hence the
power:

P = E C =
, . *
v
• f
The condition of maximum output is,
8 r
that is,
O =
(r +
^o)
8
+
V
—
2 r
(#•
+
r
0
),
or,
*2
=
r
0
2
+
s
0
2

,
r =
i/r
0
*
+
*
0
»,
and the maximum output is,
E*
P =
2
K
+
i^+V}
^
r
*
_]_
3^
at the efficiency,
>'o
+ r
r
Q
+
V
r*
-f-

*
0
*
In the instance,
E
0
=
100,
/•„
= 1,
*
0
= 10
is:
P = 453 watts, against 2,500 watts with continuous currents.
If, however, we shunt the receiver circuit by capacity induc-
tance
k,
we have,
Leads,
I
0
=
r
0
—j *
0
,
Consumer circuit, / =
r,

* = 0,
Condenser circuit,
/,
= j
k,
r
x
=
O,
hence, by substituting in the equations derived in the first part
of this chapter,
ÆTHERFORCE
8TEINMETZ ON COMPLEX QUANTITIES. 47
a =
r
0
r
-f *
0
k
b =
s
0
r
—
k(r
0
-\-
r)
or, substituting,

.
a
tan
<w
= —
-L
0
E k
and, C =
^ J
y
(cos
<5
+
.;
sin
&)
E=
Cr =
v
^ , y
t
008
* +i
8m
"0
hence, power,
P
= ^7^=
^'P*

= & & r
a
%
-\-W
(r
0
r
-+-
s
0
kf
-{-
(s
0
r —
k
r
0
— k
rf
The condition of the maximum output P is,
or
ok
that is,
#
(r
0
%
+
O

=
t*
(r
0
s
+
[«
0
-
k]*)
ks
0
=
r
0
2
+
s<?
hence,
k
=
**o
"T"
*o
*o
r
-r
0
s
+

s
0
a
substituting this in
P,
we
get:
U
P= &
the same condition as for continuous current.
That means,
*' No
matter how large the
self-induction
of an alternating
current circuit is, by a proper use of shunted capacity the out-
put of the circuit can always
be
raised to the same as for con-
tinuous
currents;
that
is,
the effect of self-induction upon the
output can
entirely,
and completely
be
annihilated^
ÆTHERFORCE

48
8TEINMETZ
ON
COMPLEX
QUANTITIES.
III. THE ALTERNATE CURRENT
TRANSFORMER.
A.
General Remarks. -
In the coils of an alternate current transformer,
E.
M.
F.
is
in-
dnced by the alternations of the magnetism, which is produced
by the combined magnetizing effect of primary and secondary
current.
If, M
—
maximum magnetism,
If
= frequency (complete cycles per second),
n
=
number of turns,
the effective intensity of the induced
E.
M.
F.

is,
F=
V~2znN
MlQr*
= 4.44 n
N
M
10"
8
Hence, if
E.
M.
F.,
frequency and number of turns are given,
or chosen, this formula gives the maximum magnetism,
f2^»
N
To produce the magnetism
M
of the transformer, a
M. M.
F.
F\%
required, which is determined from the shape and the mag-
netic characteristic of the iron, in the usual manner.
At no load, or open secondary circuit, the
M. M.
F.
F is
fur-

nished by the "exciting current," improperly called the "leakage
current?''
The energy of this current is the energy consumed by
hystere-
sis and
eddy-currents
in the
iron;
its intensity represents the
This current is not a
sine-wave,
but is
distorted
by hysteresis.
It reaches its maximum together with the maximum of magnet-
ism, but passes through zero long before the magnetism.
This exciting current can be dissolved in two components:
a
sine-wave Coo
of equal intensity and
equal
power with the
ex-
citing
current,
and a wattless complex higher harmonic.
Practically this separation is made by the
electro-dynamometer.
Connecting ammeter, voltmeter and wattmeter into the
primary of an alternate current

transformer,
at open secondary
circuit the instrument readings give the current
O^
in intensity
and phase, but suppress the higher harmonics.
In Fig. 5 such a wave is shown in rectangular coordinates.
The sine-wave of magnetism is represented by the dotted curve
M,
the exciting current by the distorted curve c, which is sepa-
rated into the
sine-wave
Co*
and the higher harmonic C.
ÆTHERFORCE
8TEINMETZ ON COMPLEX QUANTITIES
49
As seen, the higher harmonic is
email,
even in a closed circuit
transformer, compared with the exciting current
<7
00
,
and since
Coo
itself is only a few per
cent,
of the whole primary current,
the higher harmonic can for all practical purposes be suppressed.

All tests made on transformers by
electro-dynamometer
methods suppress the higher harmonic anyway.
Representing the exciting current by a sine-wave
C
o0
of equal
effective intensity and equal power with the distorted wave, the
exciting current is advanced in phase against the magnetism by
an angle a, which may be called the
"
angle of
ZiystereHc
ad-
vance of
phase"
This angle
a
is very small in all open circuit
transformers, but may be as large as 40° to 50° in closed circuit
transformers.
PIG. 5.
We can now in the usual manner dissolve the sine-wave of ex-
citing current
C^
into its two rectangular
components:
A, the
"
hysteretic energy

current"
at right angles with the
magnetism, hence in phase with the induced
E.
M.
F.,
and, there-
fore representing consumption of
energy ;
and,
g,
the
"
magnetising
current"
in phase with the magnetism,
hence at right angles with the induced E.
M.
F.,
and, therefore,
wattless.
h
=
G^
sin a, and can be calculated from the loss of energy
by hysteresis (and eddies), for it
is:
,
energy consumed by
hysteresis

primary
E.
M.
F. '
ÆTHERFORCE
50
STEINMETZ ON COMPLEX QUANTITIES.
And since
C^
can be calculated from shape and characteristic
of the iron, the angle of hysteretic advance of phase
a
is given
by:
sin a
=
A
a
00
The magnetizing current g =
C
00
cos a does not consume
energy (except by resistance), and can be supplied by a condenser
of suitable capacity shunted to the transformer.
Since in the closed circuit transformer
h,
which cannot be
supplied by a condenser, is not much smaller than
0^,

there is
no advantage in using a condenser on a closed circuit transformer.
In an open circuit transformer, however, or
transformer
motor,
Coo is
very much larger than A, and a condenser may be of ad-
vantage in reducing the exciting current from
C^
to
h.
B.—The
Closed Circuit Lighting Transformer.
The alternate current transformer with closed magnetic cir-
cuit,
when feeding into a non-inductive resistance, as lights, can
be characterized by four constants :
p = resistance loss as fraction of the total transformed power:
resistance loss
P =
at full load.
total power
e
= hysteretic loss as fraction of the total transformed power:
£ =
hysteretic loss
at full load.
total power
a
—

B.
M.
F. of self-induction as
fraction
of total E.
M.
F.
:
self-induction
a =
at full load.
total E.
M.
F.
r =
magnetizing
current as fraction of total
current:
r =
magnetizing current
total current
at full load.
Denoting
In
primary:
n
0
r
Q
C

0
K
K
we have
then:
In secondary
and
n
x
fx
C\
E>
E
t
=
number
of turns.
= resistances.
= currents.
=
induced E.
M.
F.'S.
= E.
M.
F.'S
at terminals.
= currents at full load.
C
-i\

C
x
r
ÆTHERFORCE
8TEINMBTZ
ON COMPLEX QUANTITIES.
51
„=-§,whe
re
S = magnetism leaking between primary and secondary.
M
—
magnetism snrronnding both primary and secondary.
h
T =
h
Hence at the fraction
#
of full lead.
choosing the induced E.
M.
F.
as the real axis of coordinates,
the magnetism as the imaginary axis of coordinates), we have,
Primary exciting current,
C
0O
=
h+jg
Primary current

correspond-)
Q
_
«,
Q
&
to secondary current
C
u
'
n
a
ing to secondary
v,u*i™i<
^„
hence, total primary current,
C
0
=
0'+
Co© = —
G
l
-\-h-j~j:/
»0
and, ratio of currents,
*
=
—' -f *\f &
Since, however,

0
00
= h +
j g =
0
o
H
e
+ j T) =
5L Cl»
(«
+
j
r) =
5.
C\
'+11
»o
*
we have, substituted,
Ratio of Currents.
6',
*J
'>
^ *
»
=
??2 V (l + 4
)
~f"

(
4
)
or
>*
orrae
di
um
andlargeload,
The
B.
M.
F.
at the secondary terminals is,
E
%
=E
x
-G
x
r
{
=E
x
jl-/°-f |
at the primary terminals,
E=
E
0
+

C
0
(r ;
«.)
=
E
0
| 1 +
, *
-j*
U
J-
ÆTHERFORCE
52
STEINMETZ
ON COMPLEX QUANTITIES.
hence, since
E
0
=
—5-
E
t
,
w,
Ratio of E.
M.
F.'S
atf
terminals.

Difference of phase
cu between
E.
M.
F. at primary terminals
and primary currents.
Since we have seen, that multiplying a complex quantity by
(cos
to
4*
j
B
i
n
<")>
means rotating its vector by angle
tu,
the
difference of phase between primary current and E.
M.
F., tu
ie,
given by,
G
0
= a (cos tu
-f-
j sin
w)
E

C
or, a (cos
GO
-j-
^
sin
w)
=
- ?
where
to
is the difference of phase, and a a constant.
Since in the present case the secondary current is in phase
W
with the secondary E.
M.
F.,
it is, b =
-~-
combining
this with the foregoing, we have,
C E
a b (cos
w
-f-
j sin
tu)
=
—£ _£
U:/i

hence,
_|_
p
3
—j a if
•
h
fiin
*
=
(t)'(''
+
F)
and,
Difference of phase between primary current
and
E.
M.
F. at
terminal
s.
tan to =
a
ft 4- —
T
ft
ÆTHERFORCE
STEINMETZ ON COMPLEX QUANTITIES. 58
hence:
"

With varying load
&,
the difference of phase
w
or the Jug,
first
decreases,
reaches a minimum at a d =
—
or
&
=
\/ —
,
t?
a
and afterwards increases
again."
At light loads it is mainly the magnetizing current r, at large
load the self-induction
a,
which determine the lag.
The formula, tan
ai
=
a & +
-r-
*
8 on
ty

an
approximation, and
ceases to hold for any light load, where we have to use the
complete expression.
°»
+
T
tan to —
1
—
px
&
+
J-
The efficiency is, 1 —
(p
d
+ -i
V and
tht
Loss
coefficient,
p
& -f
hence
a
minimum at,
#
=
V—,

the point of
maximum
efficiency.
p
Let, as an instance,
be:
n
0
_
10
p =
-02
e = .03
7H
a
= .06 r = .08
hence,
at full load,
& =
1,
•S
= .1 (1 + .03 + .0032) = .1033
fy = 10 (1 + .02
-f
.0018) = 10.22
tan
<u
= .06 + .08 = .14, or, at = 8°,
energy factor, cos
at

= .99
at
100$
overload,
&
= 2,
_£°
= .1 (1 + .015 +
.00U8)
= .1016
-§
= 10 (1 + .04 +
.(H»72)
= 10.47
ÆTHERFORCE
54 BTEINMETZ
ON COMPLEX QUANTITIES.
tan to = .12 + .04 = .16, or, to 9°,
energy factor, cos to = .99
at
one-half
load:
ri
#
=
.5:
-£
= .1 (1 + .06 + .0128) =
.1073.
V

-£ = 10 (1 + .01
-f
.0005) =
10.11.
tanai=.03+.16=.19,
or
«5=ll°,energy
factor:
cos<3=.98.
at
one-tenth
load:
-JJ
= 1 (1 + .3
-f-
.32) = .162.
or more exactly,
= .1
4/(1
+
.3)
2
+
.8*
= .153.
E
-g?
=
10 (1 + .002 + .0000) = 10.02.
tan

o>
= .006
-f-
.8 = .806.
or more exactly,
.006+.8
=
1
AQO
i
Q=-62,
or
<o=32°,
energy factor: cos
<5=.85,
at open
secondary:
.08
tan
tZ
=
QQ
= 2.67,
orw
= 70°, energy factor: cos
to
=
.34,
the minimum lag takes place
at:

*
=
\/'°*
=
1.155,
.0o
or
15i
per
cent,
overload, and is:
tan<J=.0693+.0693=.1386,or£j=7.9
o
,energy
factor:
cos^=.99,
the efficiency is a maximum
at:
# = \/^ =
1.225.
.02
or
22£
per
cent,
overload, and is :
1—.0245
—
.0245
=

.951,
or 95.1 per
cent
C.—Genera'.
Equations of Alternate Current Transformer.
The foregoing considerations will apply strictly only to the
closed circuit transformer, where p,
a
2
,
e,
r
2
are so
Bmall
that their
ÆTHERFORCE
STEINMETZ
ON
COMPLEX
QUANTITIES.
55
products and higher powers may be neglected when feeding into
a non-inductive resistance.
The open circuit transformer, and in general the transformer
feeding into an inductive circuit-in which case a and r become
of greatly increased
importance—requires
a fuller consideration.
Let:

n
0
and
n
x
= number of turns,
r
0
"
r
x
= resistance,
s
0
=
2TC
N
Z
0
and
s
x
= 2
it
N
L
x
= self-inductances,
hence:
I

Q
=
r
0
—j
s
0
and
l
x
=
r
x
—j
«,
= impedances of the two
transformer coils.
The secondary terminals may be connected to a circuit of re-
sistance R and inductance
S,
hence of impedance
/
= R — j
S.
Then we
have:
Magnetism:
j M.
Secondary induced E.
M.

F. :
E
x
=
VZTT
n
x
N
M10~\
Primary induced
E.
M.
F.
:
E
0
=
*
2
it
n
0
N
M10
~
8
-
E
x
.

W W
Secondary
current:
C\
=
j^
=
^-^-j-^-—
}
or:
d
=
a -{-j
b,
where:
E
x
(R +
r.)
.
E
x
(S
4
•.)
0>
= /
r> i „ \2 i /o
i—7^2,
0 —

-•-
{R +
r
x
f
+
(6'
4
*«)
2
'
""(*+
r,J»
4 (S 4
**)
8
'
Primary current corresponding hereto:
„
n
t
n
x
.
n
x
C = —
C
x
= -

a
4-
i —
o.
Primary exciting current:
^oo
=
h+J9,
hence, total primary current:
<i =
<H-ft.
=
(£«
+
*)+>•(=;»
+
,)
or:
C
0
=
o 4 i «?»
where:
^E
x
(R^r
x
)
f
_ ^ff+'O

c _
(3T+
n)
2
4
(£
4-
»,)*
+
''
~
(^
4
r,)
2
4 <S 4
*,)'
+
9t
herefrom we
get:
ÆTHERFORCE
56
8TEINMBTZ
ON
COMPLEX
QUANTITIES.
E.
M.
F.

consumed
by
primary
impedance:
C
0
1
0
=
(c+
j
d)
(r
0
—
j
g
0
)
=
{cr
0
+ d
s
0
)
+ j
(dr
0
— e

s
0
).
E.
M.
F. consumed by secondary impedance
:
C
l
I
1
=
{a+jb)(r
l
-js
1
)
=
(« ''i
+
* *i)
+ j
{br
x
—a
*,)•
hence,
E.
M.
F.

at
secondary
terminals:
7? w n
T
JP
\
1
(«n +
b8
l
)+j(br
l
— a8
x
))
IL
%
=
/S,
—
Oi
l
x
=
IL
X
<
1 —
jp

>
.
E.M.F.
at primary terminals:
'
E=
E
0
+
C.
U
=
E
\l +
<
-
er
'
+
d
'
:
>
+
J<-
dr
'-")\.
Substituting
now
in

C
u
0
o
,
E
t1
E
the values
of
a,
J,
c,
d,
we
get:
Secondary
current:
E
x
(R +
*•,)
is;
(£
+
«Q
0,
-(-ff
+
r,)

2
+
(S
+
*,)
2
+'
(£
+
r,J»
+
(5
+
s
x
f
Primary
current:
°~
((.ff+ny+C^f »i)
+
f
+ ^((7?+^+s-
K
)'
+
?
E.
M.
F.

atf
secondary terminals,
F
- F
\
i_
r
i(^+!j)+*i(^4-*i)
I
_
,• jp
i
Sr
x
—Rs
l
E.
M.
F. at primary
terminals,
*=[5s*
j
I+(
»I
Y^
^.Hi^^+i!) I
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the general equations
of the
alternate current
transformer,
representing the currents
and
E.
M.
F.'#
in
intensity and phase.
In general,
the
percentage

of
resistance
in
inductance will
be
the same, or can without noticeable error
be
assumed the same
in primary as
in
secondary circuit.
That
means,
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ÆTHERFORCE